Tuning methods for tunable matching networks

ABSTRACT

Methods for tuning a tunable matching network can involve comparing a source impedance of a source to a real part of a load impedance of a load. Depending on characteristics of the network, capacitances of one or more tunable capacitors can be set to correspond to device boundary parameters, and capacitances of remaining tunable capacitors can be set based on a predetermined relationship between the parameters of the capacitors, the source, the load, and other components. From these initially determined values, the capacitance value of one or more of the capacitors can be adjusted to fall within device boundary conditions and achieve a perfect or at least best match tuning configuration.

RELATED APPLICATIONS

The presently disclosed subject matter claims the benefit of U.S.Provisional Patent Application Ser. No. 61/401,727, filed Aug. 18, 2010,the disclosure of which is incorporated herein by reference in itsentirety.

TECHNICAL FIELD

The subject matter disclosed herein relates generally to methods foroperating electronic devices. More particularly, the subject matterdisclosed herein relates to methods for tuning a tunable matchingnetwork.

BACKGROUND

Matching networks that utilize tunable components can be useful formatching variable loads and/or optimizing performance at multiplefrequencies. It has been recognized, however, that tuning a networkcontaining multiple tunable components is not an easy job without usinga simulation optimizer. Even then, the match tuning of using such anoptimizer can be a slow process. For example, matching 300 loadimpedances might take a couple of hours using currently availablecommercial optimizers. Furthermore, the optimizer might not be able toachieve optimum results because of the tendency of iterative values tobecome trapped in local minimum or maximum values.

As a result, it would be desirable for a tuning algorithm for such atunable matching network to not only perform much faster the simulationoptimizers, but also to provide a deterministic and unique solution.

SUMMARY

In accordance with this disclosure, novel methods for tuning a tunablematching network are provided. In one aspect, a method for tuning atunable matching network can comprise a first variable capacitorcomprising a terminal connected to a first node, a second variablecapacitor comprising a terminal connected to a second node, and a firstinductor and a third variable capacitor connected in parallel betweenthe first and second nodes is provided. The method can comprisecomparing a source impedance of a source connected to the first node toa real part of a load impedance of a load connected to the second node.When the source impedance is greater than the real part of the loadimpedance or when a calculated capacitance of the second variablecapacitor is less than a minimum capacitance, a capacitance value of thesecond variable capacitor can be set to be a minimum capacitance of thesecond variable capacitor, and a capacitance value of the first variablecapacitor can be determined based on a predetermined relationshipbetween the capacitance of the first variable capacitor, a conductanceof the source, and an equivalent conductance of the first inductor andthe third variable capacitor. Alternatively, when the source impedanceis less than the real part of the load impedance or when a calculatedcapacitance of the first variable capacitor is less than a minimumcapacitance, a capacitance value of the first variable capacitor can beset to be a minimum capacitance of the first variable capacitor, and acapacitance value of the second variable capacitor can be determinedbased on a predetermined relationship between the capacitance of thesecond variable capacitor, a minimum susceptance of the first variablecapacitor, an impedance of the source, a conductance of the load, and asusceptance of the load.

Either way, the method can further comprise determining a capacitivevalue of the third variable capacitor based on a predeterminedrelationship between the capacitance of the third variable capacitor, aninductance of the first inductor, and an equivalent series inductance ofthe first inductor and the third variable capacitor, adjusting thecapacitance value of one or more of the first variable capacitor, secondvariable capacitor, or third variable capacitor if the capacitance valueis less than a minimum capacitance or greater than a maximum capacitancefor the first variable capacitor, second variable capacitor, or thirdvariable capacitor, respectively, and setting the capacitances of thefirst variable capacitor, the second variable capacitor, and the thirdvariable capacitor to be equal to the respective capacitance values.

Although some of the aspects of the subject matter disclosed herein havebeen stated hereinabove, and which are achieved in whole or in part bythe presently disclosed subject matter, other aspects will becomeevident as the description proceeds when taken in connection with theaccompanying drawings as best described hereinbelow.

BRIEF DESCRIPTION OF THE DRAWINGS

The features and advantages of the present subject matter will be morereadily understood from the following detailed description which shouldbe read in conjunction with the accompanying drawings that are givenmerely by way of explanatory and non-limiting example, and in which:

FIG. 1 a is a circuit arrangement for a tunable single section Pimatching network;

FIG. 1 b is a circuit arrangement for a tunable capacitor-bridged doublePi matching network;

FIG. 2 is a circuit arrangement for a tunable single section Pi matchingnetwork connected to a source and a load;

FIG. 3 is a flow chart illustrating a tuning method according to anembodiment of the presently disclosed subject matter;

FIG. 4 is a flow chart illustrating a tuning method according to anembodiment of the presently disclosed subject matter;

FIG. 5 is a circuit arrangement for a tunable capacitor-bridged doublePi matching network connected to a source and a load;

FIG. 6 is a flow chart illustrating a tuning method according to anembodiment of the presently disclosed subject matter;

FIGS. 7 a and 7 b are graphs illustrating input voltage standing waveratio contour plots for networks tuned using methods according to anembodiment of the presently disclosed subject matter;

FIGS. 7 c and 7 d are graphs illustrating input voltage standing waveratio contour plots for networks tuned using conventional simulationoptimizer methods;

FIGS. 8 a and 8 b are graphs illustrating relative transducer gainresults for load reflection coefficients of networks at 2170 MHz for thecase of a lossless network using methods according to an embodiment ofthe presently disclosed subject matter and conventional simulationoptimizer methods, respectively;

FIGS. 9 a and 9 b are graphs illustrating relative transducer gainresults for load reflection coefficients of networks at 700 MHz usingmethods according to an embodiment of the presently disclosed subjectmatter and conventional simulation optimizer methods, respectively;

FIGS. 10 a and 10 b are graphs illustrating relative transducer gainversus load reflection coefficients using gain results for loadreflection coefficients of networks at 700 MHz using methods accordingto an embodiment of the presently disclosed subject matter andconventional simulation optimizer methods, respectively;

FIGS. 11 a and 11 b are graphs illustrating input voltage standing waveratio improvement for a low frequency band and a high frequency band,respectively, after using methods according to an embodiment of thepresently disclosed subject matter; and

FIGS. 12 a and 12 b are graphs illustrating relative transducer gainversus frequency for a low frequency band and a high frequency band,respectively, after using methods according to an embodiment of thepresently disclosed subject matter.

DETAILED DESCRIPTION

The present subject matter provides methods for tuning a tunablematching network. It is believed that the tuning methods disclosedherein can do the same job with similar accuracy as a simulationoptimizer, but the speed of these methods can be more than 1000 timesfaster than the speed of the optimizer.

Single Pi Network Configuration

In one aspect, for example, methods according to the present subjectmatter can be applied to a tunable Pi network having a configurationshown in FIG. 1 a. A tunable Pi network, generally designated 10, cantheoretically make all load impedances over the entire Smith chartgetting conjugation match if the three tunable components in Pi network10 can be tuned to any desired value. Pi network 10 is one of thesimplest topologies, which is capable of perfectly matching the loadimpedances over the entire Smith chart. Of course, it should beunderstood, however, that the principles applied to the tuning oftunable Pi network 10 can be extended to methods of tuning other networkconfigurations. For example, FIG. 1 b shows a capacitor-bridged doublePi network, generally designated 20, which is also sometimes referred toas a bypassed lumped-TL.

In Pi network 10 shown in FIG. 1 a, a first tunable MEMS capacitor,generally designated 11, having a first capacitance C₁ is connected to afirst node 1, a second tunable MEMS capacitor, generally designated 12,having a second capacitance C₂ is connected to a second node 2, and athird tunable MEMS capacitor, generally designated 13, having a thirdcapacitance C₃ and a first inductor, generally designated 14, having afirst inductance L₁ are connected in parallel between first node 1 andsecond node 2. Using these three tunable MEMS capacitors 11, 12, and 13,the capacitances C₁, C₂, and C₃ thus serve as three variables for thesystem. To solve for an optimum configuration of the capacitances, twoequations can be derived from the condition of impedance conjugationmatch: one from the real part and another one from imaginary part of theimpedance match equation. In order to solve these two equations havingthree variables, a value can be assigned to one of the threecapacitances.

Referring to FIG. 2, Pi network 10 can comprise a source, generallydesignated 31, having a source impedance Ro connected to first node 1and a load, generally designated 32, connected to second node 2, load 32having a load resistance R_(L) and a load inductance X_(L), with thecombination of these elements defining a load impedance Z_(L) having therelationship Z_(L)=R_(L)+j X_(L). The combination of third tunable MEMScapacitor 13 and first inductor 14 can be modeled as a single equivalentelement, generally designated 15, having an equivalent series resistanceR_(e) and an equivalent series inductance L_(e).

In the case of load resistance R_(L) being less than or equal to sourceimpedance R_(o), the second capacitance C₂ can be set to a minimumsecond capacitance C_(2,min), and a second inductance B_(C2) of thesecond capacitor can be set to a value equal to ωC_(2,min). Based onthese settings, solutions for the values of first capacitance C₁ andsecond capacitance C₃ can be determined as follows:

$\begin{matrix}{C_{1} = {\frac{1}{2\;\pi\; f}\sqrt{G_{o}\left( {G_{e} - G_{o}} \right)}}} & (1) \\{{C_{3} = \frac{L_{e} - L}{\omega^{2}L_{e}L}}{and}} & (2) \\{L_{e} = {\frac{1}{2\;\pi\; f}\left( {\sqrt{R_{e}\left( {R_{o} - R_{e}} \right)} - X_{e}} \right)}} & (3)\end{matrix}$

where an equivalent series conductance G_(e) is equal to the inverse ofequivalent series resistance R_(e). and

$\begin{matrix}{R_{e} = {{\frac{G_{L}}{G_{L}^{2} + \left( {B_{L} + B_{{C\; 2},\min}} \right)^{2}}\mspace{14mu}{and}\mspace{14mu} X_{e}} = \frac{- \left( {B_{L} + B_{{C\; 2},\min}} \right)}{G_{L}^{2} + \left( {B_{L} + B_{{C\; 2},\min}} \right)^{2}}}} & (4) \\{Y_{L} = {\frac{1}{Z_{L}} = {{G_{L} + {j\; B_{L}}} = {\frac{R_{L}}{R_{L}^{2} + X_{L}^{2}} - {j\frac{X_{L}}{R_{L}^{2} + X_{L}^{2}}}}}}} & (5)\end{matrix}$

Alternatively, in the case of load resistance R_(L) being greater thansource impedance R_(o), first capacitance C₁ can be set to a minimumfirst capacitance C_(1,min), and a first susceptance B_(C1) of firstcapacitor 11 can be set to a value equal to ωC_(1,min). Using theseparameters, solutions for the values of first capacitance C₁ and secondcapacitance C₃ can be determined as follows:

$\begin{matrix}{C_{2} = {\frac{1}{2\;\pi\; f}\left( {\sqrt{{\frac{G_{L}}{R_{o}}\left( {1 + {R_{o}^{2}B_{{C\; 1},\min}^{2}}} \right)} - G_{L}^{2}} - B_{L}} \right)}} & (6) \\{{C_{3} = \frac{L_{e} - L}{\omega^{2}L_{e}L}}{and}} & (7) \\{L_{e} = {\frac{1}{2\;\pi\; f}\left( {\frac{\sqrt{{\frac{G_{L}}{R_{o}}\left( {1 + {R_{o}^{2}B_{{C\; 1},\min}^{2}}} \right)} - G_{L}^{2}}}{\frac{G_{L}}{R_{o}}\left( {1 + {R_{o}^{2}B_{{C\; 1},\min}^{2}}} \right)} + \frac{R_{o}^{2}B_{{C\; 1},\min}}{1 + {R_{o}^{2}B_{{C\; 1},\min}^{2}}}} \right)}} & (8)\end{matrix}$

A perfect conjugation match is realized if first capacitance C₁ andthird capacitance C₃ resulting from Equations (1) and (2) or secondcapacitance C₂ and third capacitance C₃ obtained from Equations (6) and(7) are within their boundary values (i.e., C_(k,min)≦C_(k)≦C_(k,max)for k=1, 2, or 3). Otherwise, further calculations can be performed forthe best match solutions based on maximizing Transducer Gain(TG)/Relative Transducer Gain (RTG) and/or minimizing voltage standingwave ratio (VSWR).

In the case of first capacitance C₁ having a value equal to a maximumfirst capacitance C_(1,max) of first capacitor 11 and second capacitanceC₂ being equal to minimum second capacitance C_(2,min) of secondcapacitor 12, or in the case of first capacitance C₁ being equal tominimum first capacitance C_(1,min) of first capacitor 11 and secondcapacitance C₂ being equal to a maximum second capacitance C_(2,max) ofsecond capacitor 12, equivalent series inductance L_(e) providing thebest match resulting from ∂VSWR_(in)/∂L_(e)=0 can be determined by thefollowing relationship:

$\begin{matrix}{{L_{e} = {\frac{1}{2\;\pi\; f} \cdot \frac{B_{C\; 2x} + B_{L} + {B_{C\; 1x}{R_{o}^{2}\left\lbrack {{\left( {B_{C\; 2x} + B_{C\; 2x} + B_{L}} \right)\left( {B_{C\; 2x} + B_{L}} \right)} + G_{L}^{2}} \right\rbrack}}}{\left\lbrack {\left( {B_{C\; 2x} + B_{L}} \right)^{2} + G_{L}^{2}} \right\rbrack \cdot \left( {{B_{C\; 1x}^{2}R_{o}^{2}} + 1} \right)}}}{where}{B_{Ckx} = {{{\omega \cdot C_{k,\min}}\mspace{14mu}{or}\mspace{14mu} C_{k,\max}\mspace{14mu}{for}\mspace{14mu} k} = {1\mspace{14mu}{or}\mspace{14mu} 2}}}} & (9)\end{matrix}$

In circumstances where third capacitance C₃ is calculated to be lessthan a minimum third capacitance C_(3,min) of third capacitor 13 orgreater than a maximum third capacitance C_(3,max) of third capacitor13, third capacitance can be set to be equal to minimum thirdcapacitance C_(3,max) or maximum third capacitance C_(3,max),respectively. If the value of second capacitance C₂ has already beenassigned to either minimum second capacitance C_(2,min) or maximumsecond capacitance C_(2,max), the value of first capacitance C₁ can bechosen to minimize the input VSWR of the Pi network tuner from∂VSWR_(in)/∂C₁=0 as follows:

$\begin{matrix}{{C_{1} = {\frac{1}{2\;\pi\; f} \cdot \frac{{X_{Le}\left\lfloor {\left( {B_{L} + B_{C\; 2x}} \right)^{2} + G_{L}^{2}} \right\rfloor} - \left( {B_{L} + B_{C\; 2x}} \right)}{\left\lbrack {1 - {X_{Le}\left( {B_{L} + B_{C\; 2x}} \right)}} \right\rbrack^{2} + {X_{Le}^{2}G_{L}^{2}}}}}{where}} & (10) \\{X_{Le} = {{\omega\; L_{e}} = {\frac{\omega\; L}{1 - {\omega^{2}{LC}_{3,\min}}}\mspace{14mu}{or}\mspace{14mu}\frac{\omega\; L}{1 - {\omega^{2}{LC}_{3,\max}}}}}} & \;\end{matrix}$

On the other hand, if first capacitance C₁ has been defined as equal tominimum first capacitance C_(1,min) or maximum first capacitanceC_(1,max), second capacitance C₂ can be determined to minimize the inputVSWR of the Pi network tuner derived from ∂VSWR_(in)/∂C₂=0 as follows:

$\begin{matrix}{C_{2} = {\frac{1}{2\;\pi\; f} \cdot \frac{\begin{matrix}{{X_{Le}\left\lbrack {1 - {X_{Le}{B_{L}\left( {1 + {R_{o}^{2}B_{C\; 1x}^{2}}} \right)}} + {B_{C\; 1x}{R_{o}^{2}\left( {{2\; B_{L}} + B_{C\; 1x}} \right)}}} \right\rbrack} -} \\{R_{o}^{2}\left( {B_{L} + B_{C\; 1x}} \right)}\end{matrix}}{X_{Le}^{2} + {R_{o}^{2}\left( {1 - {X_{Le}B_{C\; 1x}}} \right)}^{2}}}} & (11)\end{matrix}$

Based on these relationships, the method for tuning a tunable matchingnetwork can follow the steps laid out in the flow chart shown in FIG. 3.Specifically, source impedance R_(o) can be compared against loadresistance R_(L) (i.e., the real part of load impedance Z_(L)). In caseswhere source impedance R_(o) is greater than load resistance R_(L), orwhen a calculated value for second capacitance C₂ is less than minimumsecond capacitance C_(2,min), the tuning method can involve the stepsoutlined in Branch 1 of the flow chart illustrated in FIG. 3. Secondcapacitance value C₂ can be set to be equal to minimum secondcapacitance C_(2,min), and first capacitance C₁ can be determined basedon a predetermined relationship between first capacitance C₁, sourceconductance G_(o), and equivalent series conductance B_(e).Specifically, using values for load conductance G_(L) and loadsusceptance B_(L) calculated from Equation (5), equivalent loadimpedance Z_(e) can be determined using Equation (4), which can in turnbe used to determine values for first capacitance C₁ and equivalentseries inductance L_(e) using Equations (1) and (3), respectively.

If the real part of equivalent load inductance L_(e) is less than 0, orif first capacitance C₁ is less than minimum first capacitanceC_(1,min), then the tuning method can follow the steps outlined below asif source impedance R_(o) was less than load resistance R_(L), which isprovided in Branch 2 of the flow chart illustrated in FIG. 3. Otherwise,third capacitance C₃ can be determined based on the predeterminedrelationship between third capacitance C₃, first inductance L₁, andequivalent series inductance L_(e). Specifically, if first capacitanceC₁ is less than or equal to maximum first capacitance C_(1,max), thirdcapacitance C₃ using the relationship described by Equation (2). Iffirst capacitance C₁ is greater than maximum first capacitanceC_(1,max), however, first capacitance C₁ can be set to be equal tomaximum first capacitance C_(1,max), equivalent series inductance L_(e)can be re-computed using Equation (9), and third capacitance C₃ can bedetermined using Equation (2).

If these steps result in values for first capacitance C₁, secondcapacitance C₂, and third capacitance C₃ that are within the permissibleranges for each device (i.e., C_(k,min)≦C_(k)≦C_(k,max)), then a perfectmatch is achieved for tuning the system. Otherwise, further tuning stepscan be taken to achieve a best possible match. Specifically, if thirdcapacitance C₃ is less than minimum third capacitance C_(3,min), thirdcapacitance C₃ can be set to be equal to minimum third capacitanceC_(3,min), and the steps outlined in the flow chart illustrated in FIG.4 can be performed. Alternatively, if third capacitance C₃ is greaterthan maximum third capacitance C_(3,max), third capacitance C₃ can beset to be equal to maximum third capacitance C_(3,max) if thirdcapacitance C₃ is less than or equal to 1/(ω²L), or third capacitance C₃can be set to be equal to minimum third capacitance C_(3,min) if thirdcapacitance C₃ is less than or equal to 1/(ω²L). In either case, themethod can further comprise the relevant steps outlined in the flowchart illustrated in FIG. 4, which is discussed hereinbelow.

In cases where source impedance R_(o) is less than load resistanceR_(L), or the real part of equivalent load inductance L_(e) is less than0 or if first capacitance C₁ is less than minimum first capacitanceC_(1,min) as discussed above, the tuning method can involve the stepsoutlined in Branch 2 of the flow chart illustrated in FIG. 3.Specifically, first capacitance C₁ can be set to be equal to minimumfirst capacitance C_(1,min), and second capacitance C₂ can be determinedbased on the predetermined relationship between second capacitance C₂, aminimum susceptance of the first variable capacitor B_(C1,min), sourceimpedance R_(o), load conductance G_(L), and load susceptance B_(L). Inparticular, values for second capacitance C₂ and equivalent seriesinductance L_(e) can be calculated based on the relationships defined byEquations (6) and (8), respectively. If the value for second capacitanceC₂ is calculated to be less than minimum second capacitance C_(2,min),then the tuning method can be reapplied using the steps outlined inBranch 1 of FIG. 3 and discussed above.

If the value for second capacitance C₂ is determined to be less thanmaximum second capacitance C_(2,max), third capacitance C₃ can becalculated using Equation (6) from the value of equivalent seriesinductance L_(e) derived from Equation (8). Otherwise, secondcapacitance C₂ can be set to be equal to maximum second capacitanceC_(2,max), equivalent series inductance L_(e) can be recomputed usingEquation (9), and third capacitance C₃ can be calculated using Equation(6). Again, if these steps result in values for first capacitance C₁,second capacitance C₂, and third capacitance C₃ that are within thepermissible ranges for each device (i.e., C_(k,min)≦C_(k)≦C_(k,max)),then a perfect match is achieved for tuning the system.

Otherwise, if third capacitance C₃ is less than minimum thirdcapacitance C_(3,max), third capacitance C₃ can be set to be equal tominimum third capacitance C_(3,max), and the steps outlined in the flowchart illustrated in FIG. 4 can be performed. Alternatively, if thirdcapacitance C₃ is greater than maximum third capacitance C_(3,max),third capacitance C₃ can be set to be equal to maximum third capacitanceC_(3,max) if third capacitance C₃ is less than or equal to 1/(ω²L), orthird capacitance C₃ can be set to be equal to minimum third capacitanceC_(3,max) if third capacitance C₃ is less than or equal to 1/(ω²L). Ineither case, the method can further comprise the relevant steps outlinedin the flow chart illustrated in FIG. 4, which is discussed hereinbelow.

If it is determined that a perfect tuning match cannot be achieved asdiscussed above, further tuning steps can be taken to achieve a matchthat is not perfect but is the best possible match for the given systemparameters. Referring to FIG. 4, where first capacitance C₁ is set tominimum first capacitance C_(1,min) or to maximum first capacitanceC_(1,max) and third capacitance C₃ is set to minimum third capacitanceC_(3,max) or maximum third capacitance C_(3,max) for the reasonsdiscussed above, second capacitance C₂ can be calculated using Equation(11). Alternatively, where second capacitance C₂ is set to be equal tominimum second capacitance C_(2,min) or maximum second capacitanceC_(2,max) and third capacitance C₃ is set to minimum third capacitanceC_(3,min), or to maximum third capacitance C_(3,max) for the reasonsdiscussed above, first capacitance C₁ can be calculated using Equation(10) for the best match.

Once these values are determined, the best match tuning method canfurther involve searching for the maximum TG/RTG based on values offirst capacitance C₁ and second capacitance C₂ calculated from Equations(10) and (11). The RTG is calculated by using calculations involving Sparameters of the network tuner:

$\begin{matrix}{{{R\; T\; G} = \frac{{S_{21}}^{2}}{{{1 - {S_{22}\Gamma_{L}}}}^{2}}}{{where}\mspace{14mu}\Gamma_{L}\mspace{14mu}{is}\mspace{14mu}{load}\mspace{14mu}{reflection}\mspace{14mu}{coefficient}\mspace{14mu}{and}}} & (12) \\{S_{11} = \frac{{- \left( {\overset{\_}{Y_{C\; 1}} + \overset{\_}{Y_{C\; 2}}} \right)} + {\left\lfloor {1 + \left( {\overset{\_}{Y_{C\; 2}} - \overset{\_}{Y_{C\; 1}}} \right) - \overset{\_}{Y_{C\; 1}Y_{C\; 2}}} \right\rfloor \cdot \overset{\_}{Z_{Le}}}}{2 + \overset{\_}{Y_{C\; 1}} + \overset{\_}{Y_{C\; 2}} + {\left( {1 + \overset{\_}{Y_{C\; 1}} + \overset{\_}{Y_{C\; 2}} + \overset{\_}{Y_{C\; 1}Y_{C\; 2}}} \right) \cdot \overset{\_}{Z_{Le}}}}} & (13) \\{{S_{21} = {S_{12} = \frac{2}{2 + \overset{\_}{Y_{C\; 1}} + \overset{\_}{Y_{C\; 2}} + {\left( {1 + \overset{\_}{Y_{C\; 1}} + \overset{\_}{Y_{C\; 2}} + \overset{\_}{Y_{C\; 1}Y_{C\; 2}}} \right) \cdot \overset{\_}{Z_{Le}}}}}}{and}} & (14) \\{{S_{22} = \frac{{- \left( {\overset{\_}{Y_{C\; 1}} + \overset{\_}{Y_{C\; 2}}} \right)} + {\left\lbrack {1 - \left( {\overset{\_}{Y_{C\; 2}} - \overset{\_}{Y_{C\; 1}}} \right) - \overset{\_}{Y_{C\; 1}Y_{C\; 2}}} \right\rbrack \cdot \overset{\_}{Z_{Le}}}}{2 + \overset{\_}{Y_{C\; 1}} + \overset{\_}{Y_{C\; 2}} + {\left( {1 + \overset{\_}{Y_{C\; 1}} + \overset{\_}{Y_{C\; 2}} + \overset{\_}{Y_{C\; 1}Y_{C\; 2}}} \right) \cdot \overset{\_}{Z_{Le}}}}}{where}} & (15) \\{{\overset{\_}{Z_{Le}} = \frac{Z_{Le}}{R_{o}}},{\overset{\_}{Y_{C\; 1}} = {Y_{C\; 1} \cdot R_{o}}},{{{and}\mspace{14mu}\overset{\_}{Y_{C\; 2}}} = {Y_{C\; 2} \cdot R_{o}}}} & \; \\{Z_{Le} = {2\;\pi\;{f \cdot {L_{e}\left( {\frac{1}{Q_{Le}(f)} + j} \right)}}}} & (16) \\{{Y_{C\; 1} = {2\;\pi\;{f \cdot {C_{1}\left( {\frac{1}{Q_{C\; 1}(f)} + j} \right)}}}}{and}} & (17) \\{Y_{C\; 2} = {2\;\pi\;{f \cdot {C_{2}\left( {\frac{1}{Q_{C\; 2}(f)} + j} \right)}}}} & (18)\end{matrix}$

For the case where first capacitance C₁ is set to minimum firstcapacitance C_(1,min), and second capacitance C₂ is within the range ofC_(2,min)≦C₂≦C_(2,max), RTG= RTG₁ can be calculated using Equation (12).If second capacitance C₂ is not within the range ofC_(2,min)≦C₂≦C_(2,max), however, second capacitance C₂ can be set tominimum second capacitance C_(2,min) or to maximum second capacitanceC_(2,max) if second capacitance C₂ is less than minimum secondcapacitance C_(2,min) or greater than maximum second capacitanceC_(2,max), respectively, and RTG= RTG₁ can be calculated using Equation(12).

For the case where first capacitance C₁ is set to maximum firstcapacitance C_(1,max), and second capacitance C₂ is within the range ofC_(2,min)≦C₂≦C_(2,max), RTG= RTG₂ can be calculated using Equation (12).If second capacitance C₂ is not within the range ofC_(2,min)≦C₂≦C_(2,max), however, second capacitance C₂ can be set tominimum second capacitance C_(2,min) or to maximum second capacitanceC_(2,max) if second capacitance C₂ is less than minimum secondcapacitance C_(2,min) or greater than maximum second capacitanceC_(2,max), respectively, and RTG= RTG₂ , can be calculated usingEquation (12).

For the case where second capacitance C₂ is set to minimum secondcapacitance C_(2,min), and first capacitance C₁ is within the range ofC_(1,min)≦C₁≦C_(1,max), RTG= RTG₃ can be calculated using Equation (12).If first capacitance C₁ is within the range of C_(1,min)≦C₁≦C_(1,max),however, first capacitance C₁ can be set to minimum first capacitanceC_(1,min) or to maximum first capacitance C_(1,max) if first capacitanceC₁ is less than minimum first capacitance C_(1,min) or greater thanmaximum first capacitance C_(1,max), respectively, and RTG= RTG₃ can becomputed using Equation (12).

For the case where second capacitance C₂ is set to maximum secondcapacitance C_(2,max), and first capacitance C₁ is within the range ofC_(1,min)≦C₁≦C_(1,max), RTG= RTG₄ can be calculated using Equation (12).If first capacitance C₁ is within the range of C_(1,min)≦C₁≦C_(1,max),however, first capacitance C₁ can be set to minimum first capacitanceC_(1,min) or to maximum first capacitance C_(1,max) if first capacitanceC₁ is less than minimum first capacitance C_(1,min) or greater thanmaximum first capacitance C_(1,max), respectively, and RTG= RTG₂ can becomputed using Equation (12).

Comparing RTG₁/ RTG₁ , RTG₂/ RTG₂ , RTG₃/ RTG₃ , and RTG₄/ RTG₄ , thesolutions or the setting of the tunable capacitors can be selectedcorresponding to the largest RTG_(X) or RTG_(X) among the four RTGvalues calculated. The solutions can thus be one of the followingpermutations sets: (C_(1,min)/C_(1,max), C₂, C_(3,min)/C_(3,max)) or(C₁, C_(2,min)/C_(2,max), C_(3,min)/C_(3,max)) or (C_(1,min)/C_(1,max),C_(2,min)/C_(2,max), C_(3,min)/C_(3,max)), where it is understood thatC_(x)/C_(y) is read as C_(x) or C_(y). These are the best matchsolutions that maximize the RTG.

Capacitor-Bridged Double Pi Network Configuration

Although the above discussion involved tuning methods particularlydesigned for use with a Pi network 10, the tuning methods according tothe present subject matter can be applied to other networkconfigurations. Specifically, for example, for the capacitor-bridgeddouble Pi network 20 shown in FIG. 1 b, a similar method can be applied.Similar to Pi network 10 discussed above, capacitor-bridged double Pinetwork 20 can include a first tunable MEMS capacitor, generallydesignated 21, having a first capacitance C_(A) and connected to firstnode 1, a second tunable MEMS capacitor, generally designated 22, havinga second capacitance C_(B) connected to second node 2, and a thirdtunable MEMS capacitor, generally designated 23, having a thirdcapacitance C_(D) and a first inductor, generally designated 34, havinga first inductance L₁ connected in parallel between first node 1 andsecond node 2.

This system can differ from Pi network 10 discussed above, however, inthat it can have a second inductor, generally designated 25, having asecond inductance L₂ arranged in series with first inductor 24 betweenfirst and second nodes 1 and 2 and a fourth variable capacitor,generally designated 26, defining a fourth capacitance C_(C) connectedto a third node 3 between first inductor 24 and second inductor 25. Thissystem can be modeled as shown in FIG. 5 as a single Pi network, withfourth variable capacitor 26 being modeled as two separate elements, afirst equivalent capacitor, generally designated 27, connected betweenfirst capacitor 21 and first inductor 24 and defining a first equivalentcapacitance C_(o1) and a second equivalent capacitor, generallydesignated 28, connected between second inductor 25 and second capacitor22 and defining a second equivalent capacitance C_(o2), and with firstinductor 24 and second inductor 25 being treated as a single equivalentinductor, generally designated 29, defining an equivalent inductanceL_(o). In this way capacitor-bridged double Pi network 20 can beanalyzed as if it were a single Pi network, thereby allowing forapplication of the equations identified above by utilizing the followingformulas:

$\begin{matrix}{C_{1} = {{C_{A} + C_{o\; 1}} = {C_{A} + \frac{C_{C}}{1 + {L_{1}/L_{2}} - {\omega^{2}L_{1}C_{C}}}}}} & (19) \\{C_{2} = {{C_{B} + C_{o\; 2}} = {C_{B} + \frac{C_{C}}{1 + {L_{2}/L_{1}} - {\omega^{2}L_{2}C_{C}}}}}} & (20) \\{{C_{3} = C_{D}}{and}} & (21) \\{L_{o} = {L_{1} + L_{2} - {\omega^{2}L_{1}L_{2}C_{C}}}} & (22)\end{matrix}$

It should be recognized that these formulas differ from thoseestablished with respect to the configuration of Pi network 10 with theaddition of fourth capacitance C_(C). This means that the formulasdeveloped for use with Pi network 10 and the tuning method describedabove can be applied equally to this configuration as long as the valuefor fourth capacitance C_(C) is defined. It is noted, however, that thevalue for fourth capacitance C_(C) in the tunable matching networkoptimal match tuning goes to either a minimum fourth capacitanceC_(C,min) or a maximum fourth capacitance C_(C,max) at a probability ofaround 80% or more as discussed below. Based on a distribution of thevalue of fourth capacitance C_(C) over a certain area of the Smithchart, such as 0.5≦|Γ_(L)|≦0.9 at any angle, the tuning method can thuscomprise the following steps.

In a first step, for a load reflection coefficient in the region ofphase within −θ₁∠Γ_(L)≦+θ₂ (e.g., −70°≦∠Γ_(L)≦+90°), and magnitudewithin 0.5≦|Γ_(L)|≦0.9, fourth capacitance C_(C) can be set to minimumfourth capacitance C_(C,min), and values for first, second, and thirdcapacitances C₁ through C₃ can be determined using Equations (19)through (21). The tuning method described above with reference to theflow chart shown in FIG. 3 can be followed to process match tuningcalculations. In the RTG calculation of the Pi network tuning algorithm,all the components in the Pi network are related to the components inthe tunable matching network by Equations (19) through (22), and valuesfor each of the parameters of capacitor-bridged double Pi network 20 canbe determined based on the following relationships (assuming a finite Qfactor):

$\begin{matrix}{{\hat{C}}_{A} = {C_{A}\left( {1 - \frac{j}{Q_{CA}(f)}} \right)}} & (22) \\{{\hat{C}}_{C} = {C_{C}\left( {1 - \frac{j}{Q_{Cc}(f)}} \right)}} & (23) \\{{\hat{C}}_{E} = {C_{E}\left( {1 - \frac{j}{Q_{CE}(f)}} \right)}} & (24) \\{{{\hat{C}}_{F} = {C_{F}\left( {1 - \frac{j}{Q_{CF}(f)}} \right)}}{And}} & (25) \\{{{\hat{L}}_{k} = {L_{k}\left( {1 - \frac{j}{Q_{Lk}(f)}} \right)}}{k = {1\mspace{14mu}{and}\mspace{14mu} 2}}} & (26)\end{matrix}$

In a second step, for a load reflection coefficient in the region ofphase within +180°≦∠Γ_(L)≦+180°−θ₃ (and within −180°≦∠Γ_(L)≦−180°+θ₄),and magnitude within 0.5≦|Γ_(L)|≦0.9, the method can comprise thefollowing steps. Fourth capacitance C_(C) can be set to be equal tomaximum fourth capacitance C_(C,max), and values for first, second, andthird capacitances C₁ through C₃ can be determined using Equations (19)through (21). The tuning method disclosed above with reference to the Pinetwork can be used to obtain the maximized RTG₁. Fourth capacitanceC_(C) can then be set to be equal to a value within the range ofC_(C,min)≦C_(C)≦C_(C,max) (e.g., C_(C)=C_(C,mean)), and values forfirst, second, and third capacitances C₁ through C₃ can be again becalculated. The tuning method for the Pi network can again be used toobtain the maximized RTG₂. Values for RTG₁ and RTG₂ can be compared, andthe solution of capacitance values having the larger RIG value can beselected.

In a third step, for a load reflection coefficient having a phase withinthe rest of region ∠Γ_(L) and a magnitude within 0.5≦|Γ_(L)|≦0.9, themethod can comprise the following steps. Fourth capacitance C_(C) can beset to be equal to maximum fourth capacitance C_(C,max), and values forfirst, second, and third capacitances C₁ through C₃ can be determinedusing Equations (19) through (21). The tuning method disclosed abovewith reference to the Pi network can be used to obtain the maximizedRTG₁. Fourth capacitance C_(C) can then be set to be equal to an averagevalue between the boundary conditions of the fourth capacitor (i.e.,C_(C)=C_(C,mean)), and values for first, second, and third capacitancesC₁ through C₃ can again be calculated. The tuning method for the Pinetwork can again be used to obtain another maximized RTG₂. Fourthcapacitance C_(C) can be set to be equal to minimum fourth capacitanceC_(C,min), and values for first, second, and third capacitances C₁through C₃ can be calculated. The tuning method for the Pi network canthen be used to obtain a third maximized RTG₃. Values for RTG₁, RTG₂ andRTG₃, can be compared, and the solution of capacitance values having thelarger RTG value can be selected.

Separating the calculations for load reflection coefficients withindifferent phases can help to reduce the computation time but is notnecessary. For example, optimal simulations across 700 MHz to 2700 MHzcan show that in the middle section of −180°≦∠Γ_(L)≦−180°, the value offourth capacitance C_(c) is generally always equivalent to minimumfourth capacitance C_(c,min), and the value of fourth capacitance C_(c)is equal to maximum fourth capacitance C_(c,max) when the reflectioncoefficient angle is close to + or −180°. There is no a criterion forchoosing the θs, so long as θ_(i) (i=1, 2, 3, or 4) is not too large.

In yet another alternative approach, the calculation of capacitancevalues for load reflection coefficients in the region of phase within+180°≧∠Γ_(L)≧+180°−θ₃ can be bypassed. In this case, the load reflectioncoefficient region for the third step can be for a phase within+180°≧∠Γ_(L)≧+180°−θ₂, within −180°≦∠Γ_(L)≦−180°+θ₁, and having amagnitude within 0.5≦|Γ_(L)|≦0.9. The third step can thus be used todetermine the perfect or the best match solution of capacitance setC_(A), C_(B), C_(C), & C_(D) values with the largest RTG over the entirearea of the load reflection coefficient defined in the Smith chart, forexample for a phase within −180°≦∠Γ_(L)≦+180° and a magnitude within0.5≦|Γ_(L)|≦0.9. It is to be understood, however, that more computationtime can be required if this calculation approach is used.

Tuning Applications

In one common configuration, a Pi network tuner (e.g., configurationshown in FIG. 1 a) can have the following parameters: first and secondtunable capacitors 11 and 12 having a tuning range from 0.8 pF to 5 pF,third tunable capacitor 13 possessing a tuning range from 0.25 pF to 4pF, and first inductor 14 of 6.8 nH and 2.3 nH being used for lowfrequency band (e.g., 700 to 960 MHz) and for high frequency band (e.g.,1710 to 2170 MHz), respectively. The input VSWR contours of match tuningthe load with reflection |Γ_(L)| can vary from 0.05 to 0.95 at 700 MHzby using the tuning method disclosed above and utilizing the optimizerare shown in FIGS. 7 a and 7 c, respectively. The average VSWR over theSmith chart within the region of 0.05≦∠Γ_(L)≦0.95 and −180°≦∠Γ_(L)≦180°can be 2.15 and 2.13 for the algorithm and optimizer, respectively, andthe difference can be only Δ=0.02. The average VSWR over the Smith chartwithin the same region at 2170 MHz can be 1.212 and 1.209 resulting fromthe algorithm and the optimizer, respectively, as depicted in FIGS. 7 band 7 d.

In the region of |Γ_(L)|<0.5, the input VSWR can be very close to 1:1.In most cellular handset applications, a VSWR less than 3:1 is usuallyrequired after match tuning. Therefore, the most interesting area in theSmith chart to check the tuner functioning is within the region of0.55≦|Γ_(L)|≦0.90 and −180°≦∠Γ_(L)≦180°. Thus, the following discussionrelates to the matching performance within this region. A comparison ofthe average input VSWR obtained from the optimizer and algorithm atdifferent frequencies are given in Table 1.

TABLE 1 Optimizer Frequency Average Algorithm (MHz) VSWR Average VSWRΔVSWR 700 3.22 3.22 0.00 960 2.31 2.31 0.00 1710 1.22 1.23 0.01 21701.41 1.42 0.01

The plots of the RTG versus the load reflection coefficient (e.g.,0.5≦|Γ_(L)|≦0.9 and −180°≦∠Γ_(L)≦180°) at 2170 MHz derived from thetuning algorithm and the optimizer simulation are given in FIGS. 8 a and8 c, respectively. The average RTG over the above area is found to be3.19 dB for the algorithm and 3.20 dB for the optimizer and thedifference is only 0.01 dB. A comparison of the average RTG resultingfrom the algorithm and the optimizer at different operating frequenciesis presented in Table 2.

TABLE 2 Algorithm Frequency Optimizer Average Average RTG (MHz) RTG (dB)(dB) ΔRTG (dB) 700 2.12 2.12 0.00 960 2.68 2.67 −0.01 1710 3.31 3.30−0.01 2170 3.20 3.19 −0.01

In the practical case, all the components of Pi network 10 can have afinite Q factor instead of infinite. Accordingly, the following providesa comparison of results achieved by the present tuning methods of a Pinetwork tuner with loss with those resulting from optimizer simulations.Assuming that the tuner is formed by the components with same tunablerange and nominal value as defined in the previous example but having afinite Q factor, their quality factors are Q_(C1,2)=100 for first andsecond tunable capacitors 11 and 12, Q_(C3)=150 for third tunablecapacitor 13 and Q_(L)=55 for first inductor 14. In order to take thefinite Q of the components into account, the final RTG and/or input VSWRcalculations can use Equations (16) through (18). In the case of thetuner with loss, the plots of the RTG versus the load reflectioncoefficient (0.5≦|Γ_(L)|≦0.9 and −180°≦∠Γ_(L)≦180°) at 700 MHz derivedfrom the tuning algorithm and the optimizer simulation are given in FIG.8. The average RTG over the above area can be found to be about 1.79 dBand 1.78 dB for the optimizer and algorithm, respectively, and thedifference can be only about 0.01 dB. A comparison of the average RTGresulting from the algorithm and the optimizer at different operatingfrequencies is presented in Table 3.

TABLE 3 Algorithm Frequency Optimizer Average Average RTG (MHz) RTG (dB)(dB) ΔRTG (dB) 700 1.79 1.78 −0.01 960 2.10 2.05 −0.05 1710 2.29 2.22−0.07 2170 2.29 2.22 −0.07

The tuning method for capacitor-bridged double Pi network 20 can alsoprovide very accurate results compared with the results obtained from anMWO simulation optimizer. For example, capacitor-bridged double Pinetwork 20 shown in FIG. 1 b can have first and second inductors 24 and25 configured to have a combined inductance L_(o) that is substantiallyequivalent to the inductance L₁ of first inductor 14 of Pi network 10and having values of about 3.4 nH for 700-960 MHz, 1.5 nH for 1710-2170MHz, and 1.0 nH for 2500-2700 MHz, first and second capacitors 21 and 22having minimum capacitances C_(a,min) and C_(b,min) equal to about 1.5pF (parasitics included) and maximum capacitances C_(a,max) andC_(b,max) equal to about 6 pF, third capacitor 23 having a minimumcapacitance C_(c,min) of about 0.6 pF and a maximum capacitanceC_(c,max) of about 4.0 pF, fourth capacitor 26 having minimumcapacitance C_(d,min) of about 0.4 pF and a maximum capacitanceC_(d,max) of about 4.0 pF, and quality factors of Q_(ca)=Q_(cb)=100,Q_(cc)=Q_(cd)=150, and Q_(L)=55.

Plots of the RTG vs. load reflection coefficient at 2500 MHz resultingfrom the algorithm and the simulation optimizer are shown in FIG. 10. Inthis configuration, the difference of the average RTG obtained from thealgorithm and the optimizer can be only about 0.05 dB. A comparison ofthe average RTG derived from the algorithm and the optimizer at otherfrequencies is presented in Table 4.

TABLE 4 Optimizer Algorithm ΔRTG = Frequency Average RTG Average RTGRTG_(avg)_A − (MHz) (dB) (dB) RTG_(avg)_O (dB) 700 1.85 1.83 −0.02 8242.25 2.21 −0.04 960 2.29 2.23 −0.06 1710 2.22 2.16 −0.06 1980 2.10 2.05−0.05 2170 1.85 1.81 −0.04 2500 1.99 1.94 −0.05 2700 1.80 1.76 −0.04

In order to demonstrate the performance of the capacitor-bridged doublePi tuning algorithm, an example for matching a handset antenna isadopted here. The reflection coefficient data measured for a particularhandset antenna, for example, can be used in this study. The presentmethod can be used to maximize the RTG for each frequency point in twodifferent frequency bands: low band from 700 MHz to 960 MHz and highband from 1710 MHz to 2170 MHz. Exemplary data for the VSWR of theoriginal antenna without TMN and the VSWR after using the TMN withmaximizing RTG tuning for the low and high frequency bands are depictedin FIGS. 11 a and 11 b, respectively. From these two plots, it can beseen that the improvement of the input VSWR after using the TMN isclearly very significant if the tuning is performed at each frequency.It is noted that the antenna used in this example appears to have beendesigned for the ECell and PCS bands since in these two bands it has thelower VSWR. Therefore, its performance in 700 MHz band is very bad(e.g., the VSWR up to 43:1).

The RTG verses frequency resulting from the TMN algorithm tuning in thelow and high bands is shown in FIGS. 12 a and 12 b, respectively. It isexpected that large RTG can be obtained in the frequency region withhigh VSWR and small RTG where the VSWR is low. In fact, the RTG can alsodepend on where the load impedance locates in the Smith chart.

Accordingly, the tuning methods disclosed herein can do the same jobwith similar accuracy as the simulation optimizer, but the speed ofthese methods can be more than 1000 times faster than the speed of theoptimizer.

The present subject matter can be embodied in other forms withoutdeparture from the spirit and essential characteristics thereof. Theembodiments described therefore are to be considered in all respects asillustrative and not restrictive. Although the present subject matterhas been described in terms of certain preferred embodiments, otherembodiments that are apparent to those of ordinary skill in the art arealso within the scope of the present subject matter.

What is claimed is:
 1. A method for tuning a tunable matching networkcomprising a first variable capacitor comprising a terminal connected toa first node, a second variable capacitor comprising a terminalconnected to a second node, and a first inductor and a third variablecapacitor connected in parallel between the first and second nodes, themethod comprising: (a) comparing a source impedance of a sourceconnected to the first node to a real part of a load impedance of a loadconnected to the second node; (b) when the source impedance is greaterthan the real part of the load impedance or when a calculatedcapacitance of the second variable capacitor is less than a minimumcapacitance: (i) setting a capacitance value of the second variablecapacitor to be a minimum capacitance of the second variable capacitor;and (ii) determining a capacitance value of the first variable capacitorbased on a predetermined relationship between the capacitance of thefirst variable capacitor, a conductance of the source, and an equivalentconductance of the first inductor and the third variable capacitor; (c)when the source impedance is less than the real part of the loadimpedance or when a calculated capacitance of the first variablecapacitor is less than a minimum capacitance: (i) setting a capacitancevalue of the first variable capacitor to be a minimum capacitance of thefirst variable capacitor; and (ii) determining a capacitance value ofthe second variable capacitor based on a predetermined relationshipbetween the capacitance of the second variable capacitor, a minimumsusceptance of the first variable capacitor, an impedance of the source,a conductance of the load, and a susceptance of the load; (d)determining a capacitive value of the third variable capacitor based ona predetermined relationship between the capacitance of the thirdvariable capacitor, an inductance of the first inductor, and anequivalent series inductance of the first inductor and the thirdvariable capacitor; (e) adjusting the capacitance value of one or moreof the first variable capacitor, second variable capacitor, or thirdvariable capacitor if the capacitance value is less than a minimumcapacitance or greater than a maximum capacitance for the first variablecapacitor, second variable capacitor, or third variable capacitor,respectively; and (f) setting the capacitances of the first variablecapacitor, the second variable capacitor, and the third variablecapacitor to be equal to the respective capacitance values.
 2. Themethod of claim 1, wherein the predetermined relationship between thecapacitance of the first variable capacitor, a conductance of thesource, and an equivalent conductance of the first inductor and thethird variable capacitor comprises the relationship:$C_{1} = {\frac{1}{2\;\pi\; f}\sqrt{G_{o}\left( {G_{e} - G_{o}} \right)}}$where C₁ is the capacitance of the first variable capacitor, G₀ is theconductance of the source, and G_(e) is the equivalent conductance ofthe first inductor and the third variable capacitor.
 3. The method ofclaim 1, wherein the predetermined relationship between the capacitanceof the second variable capacitor, a minimum susceptance of the firstvariable capacitor, an impedance of the source, a conductance of theload, and a susceptance of the load comprises the relationship:$C_{2} = {\frac{1}{2\;\pi\; f}\left( {\sqrt{{\frac{G_{L}}{R_{o}}\left( {1 + {R_{o}^{2}B_{{C\; 1},\min}^{2}}} \right)} - G_{L}^{2}} - B_{L}} \right)}$where C₂ is the capacitance of the second variable capacitor, B_(C1,min)is the minimum susceptance of the first variable capacitor, R₀ is theimpedance of the source, G_(L) is the conductance of the load, and B_(L)is the susceptance of the load.
 4. The method of claim 1, wherein thepredetermined relationship between the capacitance of the third variablecapacitor, an inductance of the first inductor, and an equivalent seriesinductance of the first inductor and the third variable capacitorcomprises the relationship: $C_{3} = \frac{L_{e} - L}{\omega^{2}L_{e}L}$where C₃ is the capacitance of the third variable capacitor, L is theinductance of the first inductor, and L_(e) is the equivalent seriesinductance of the first inductor and the third variable capacitor. 5.The method of claim 1, wherein the tunable matching network furthercomprises a second inductor arranged in series with the first inductorbetween the first and second nodes and a fourth variable capacitorcomprising a terminal connected to a third node between the firstinductor and the second inductor; and wherein the method furthercomprises: (g) when a reflection coefficient of the load is within achosen region of phase: (i) setting a capacitance value of the fourthvariable capacitor to be a minimum capacitance of the fourth variablecapacitor; (ii) determining capacitance values of the first, second, andthird variable capacitors according to steps (a) through (e); and (iii)setting the capacitances of the first variable capacitor, the secondvariable capacitor, the third variable capacitor, and the fourthvariable capacitor to be equal to respective capacitance values; and (h)when a reflection coefficient of the load is outside the chose region ofphase: (i) setting a capacitance value of the fourth variable capacitorto be a maximum capacitance of the fourth variable capacitor,determining capacitance values of the first, second, and third variablecapacitors according to steps (a) through (e), and determining a firstmaximum relative transducer gain value based on the capacitance values;(ii) setting a capacitance value of the fourth variable capacitor to bea predetermined capacitance between a maximum capacitance and a minimumcapacitance of the fourth variable capacitor, determining capacitancevalues of the first, second, and third variable capacitors according tosteps (a) through (e), and determining a second maximum relativetransducer gain value based on the capacitance values; (iii) setting acapacitance value of the fourth variable capacitor to be a minimumcapacitance of the fourth variable capacitor, determining capacitancevalues of the first, second, and third variable capacitors according tosteps (a) through (e), and determining a third maximum relativetransducer gain value based on the capacitance values; and (iv) settingthe capacitances of the first variable capacitor, the second variablecapacitor, the third variable capacitor, and the fourth variablecapacitor to be equal to respective capacitance values corresponding toa highest of the first, second, or third maximum relative transducergain values.
 6. The method of claim 1, wherein adjusting the capacitancevalue of one or more of the first variable capacitor, second variablecapacitor, or third variable capacitor comprises setting the capacitancevalue of the first variable capacitor, second variable capacitor, orthird variable capacitor, respectively, to be equal to the minimumcapacitance if the capacitance value determined is less than the minimumcapacitance or to be equal to the maximum capacitance if the capacitancevalue determined is greater than the maximum capacitance.
 7. The methodof claim 6, wherein if the capacitance values of both the secondvariable capacitor and the third variable capacitor are equal to eitherthe minimum capacitance or the maximum capacitance of the secondvariable capacitor and the third variable capacitor, respectively,adjusting the capacitance value of one or more of the first variablecapacitor, second variable capacitor, or third variable capacitorfurther comprises determining a new capacitance value of the firstvariable capacitor based on a predetermined relationship between thecapacitance of the first variable capacitor, a conductance of the load,a susceptance of the load, a susceptance of the second variablecapacitor, and an equivalent series reactance of the first inductor andthe third variable capacitor.
 8. The method of claim 6, wherein if thecapacitance values of both the first variable capacitor and the thirdvariable capacitor are equal to either the minimum capacitance or themaximum capacitance of the first variable capacitor and the thirdvariable capacitor, respectively, adjusting the capacitance value of oneor more of the first variable capacitor, second variable capacitor, orthird variable capacitor further comprises determining a new capacitancevalue of the second variable capacitor based on a predeterminedrelationship between the capacitance of the second variable capacitor,an impedance of the source, a susceptance of the load, a susceptance ofthe first variable capacitor, and an equivalent series reactance of thefirst inductor and the third variable capacitor.
 9. The method of claim8, wherein the predetermined relationship between the capacitance of thesecond variable capacitor, an impedance of the source, a susceptance ofthe load, a susceptance of the first variable capacitor, and anequivalent series reactance of the first inductor and the third variablecapacitor comprises the relationship:$C_{2} = {\frac{1}{2\;\pi\; f} \cdot \frac{\begin{matrix}{{X_{Le}\left\lbrack {1 - {X_{Le}{B_{L}\left( {1 + {R_{o}^{2}B_{C\; 1x}^{2}}} \right)}} + {B_{C\; 1x}{R_{o}^{2}\left( {{2\; B_{L}} + B_{C\; 1x}} \right)}}} \right\rbrack} -} \\{R_{o}^{2}\left( {B_{L} + B_{C\; 1x}} \right)}\end{matrix}}{X_{Le}^{2} + {R_{o}^{2}\left( {1 - {X_{Le}B_{C\; 1x}}} \right)}^{2}}}$where C₂ is the capacitance of the second variable capacitor, R₀ is theimpedance of the source, B_(L) is the susceptance of the load, B_(C1x)is the susceptance of the first variable capacitor, and X_(Le) is theequivalent series reactance of the first inductor and the third variablecapacitor.
 10. The method of claim 7, wherein the predeterminedrelationship between the capacitance of the first variable capacitor, aconductance of the load, a susceptance of the load, a susceptance of thesecond variable capacitor, and an equivalent series reactance of thefirst inductor and the third variable capacitor comprises therelationship:$C_{1} = {\frac{1}{2\;\pi\; f} \cdot \frac{{X_{Le}\left\lbrack {\left( {B_{L} + B_{C\; 2x}} \right)^{2} + G_{L}^{2}} \right\rbrack} - \left( {B_{L} + B_{C\; 2x}} \right)}{\left\lbrack {1 - {X_{Le}\left( {B_{L} + B_{C\; 2x}} \right)}} \right\rbrack^{2} + {X_{Le}^{2}G_{L}^{2}}}}$where C₁ is the capacitance of the first variable capacitor, G_(L) isthe conductance of the load, B_(L) is the susceptance of the load,B_(C2x) is the susceptance of the second variable capacitor, and X_(Le)is the equivalent series reactance of the first inductor and the thirdvariable capacitor.